NCERT Solutions for Class 10 Maths Chapter 1 – Real Numbers Exercise 1.4

NCERT Solutions for Class 10 Maths Chapter 1 – Real Numbers Exercise 1.4 are provided here to help students with their Class 10 exam preparations. These solutions are designed by subject experts to explain Euclid’s Division Algorithm, gcd (Greatest Common Divisor), and applications in a detailed and step-by-step manner. The solutions are aimed at simplifying number theory concepts for students.

Exercise 1.4 in Real Numbers focuses on applying Euclid's Division Algorithm to solve more complex problems related to finding the gcd of two numbers and understanding divisibility in a practical context. By practicing this exercise, students will get a better understanding of divisibility rules, quotients, and remainders, which are fundamental concepts in number theory.

NCERT Solutions for Class 10 Maths Chapter 1 – Real Numbers Exercise 1.4

Real Numbers

Medium

Q.

$\begin{array}{l}\text{Without actually performing the long division},\\ \text{state whether the following rational\hspace{0.17em}\hspace{0.17em}numbers}\\ \text{will have a terminating decimal expansion or}\\ \text{a non}-\text{terminating repeating decimal\hspace{0.17em}\hspace{0.17em}expansion}:\\ \left(\text{i}\right)\frac{\text{13}}{\text{3125}}\left(\text{ii}\right)\frac{\text{17}}{\text{8}}\left(\text{iii}\right)\frac{\text{64}}{\text{455}}\left(\text{iv}\right)\frac{\text{15}}{\text{16}00}\\ \left(\text{v}\right)\frac{\text{29}}{\text{343}}\left(\text{vi}\right)\frac{\text{23}}{2^3{5}^2}\left(\text{vii}\right)\frac{\text{129}}{{\text{2}}^{\text{2}}{\text{5}}^7{7}^5}\left(\text{viii}\right)\frac{\text{6}}{\text{15}}\\ \left(\text{ix}\right)\frac{\text{35}}{\text{5}0}\left(\text{x}\right)\frac{\text{77}}{\text{21}0}\end{array}$

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Real Numbers

Medium

Q.

Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.

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Real Numbers

Medium

Q.

$\begin{array}{l}\text{The following real numbers have decimal expansions}\\ \text{as given below. In each case decide whether they are}\\ \text{rational or not. If they are rational, and of the form }\frac{\mathrm{p}}{\mathrm{q}},\\ \text{what can you say about the prime factors of }\mathrm{q}?\\ \\ \text{(i) 43.123456789 (ii) 0.120120012000120000}...\\ \text{(iii) 43.}\overline{123456789}\end{array}$

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These solutions are aligned with the latest CBSE syllabus, offering clear, step-by-step guidance and detailed solutions to help students perform well in board examinations.

1. Question:
   Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:
   (i)

   $\frac{13}{3125}$312513​
   (ii)

   $\frac{17}{8}$817​
   (iii)

   $\frac{64}{455}$45564​
   (iv)

   $\frac{1500}{1600}$16001500​
   (v)

   $\frac{29}{343}$34329​
   (vi)

   $\frac{23}{23^2 \cdot 5^2}$232⋅5223​
   (vii)

   $\frac{129}{2^2 \cdot 5^7}$22⋅57129​
   (viii)

   $\frac{6}{15}$156​
   (ix)

   $\frac{35}{50}$5035​
   (x)

   $\frac{77}{210}$21077​

   Answer:

   • (i)

     $\frac{13}{3125}$312513​ has a terminating decimal expansion.

   • (ii)

     $\frac{17}{8}$817​ has a terminating decimal expansion.

   • (iii)

     $\frac{64}{455}$45564​ has a non-terminating repeating decimal expansion.

   • (iv)

     $\frac{1500}{1600}$16001500​ has a terminating decimal expansion.

   • (v)

     $\frac{29}{343}$34329​ has a non-terminating repeating decimal expansion.

   • (vi)

     $\frac{23}{23^2 \cdot 5^2}$232⋅5223​ has a terminating decimal expansion.

   • (vii)

     $\frac{129}{2^2 \cdot 5^7}$22⋅57129​ has a terminating decimal expansion.

   • (viii)

     $\frac{6}{15}$156​ has a terminating decimal expansion.

   • (ix)

     $\frac{35}{50}$5035​ has a terminating decimal expansion.

   • (x)

     $\frac{77}{210}$21077​ has a non-terminating repeating decimal expansion.

   View Solution

2. Question:
   Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.

   Answer:

   • (i)

     $\frac{13}{3125} = 0.00416$312513​=0.00416

   • (ii)

     $\frac{17}{8} = 2.125$817​=2.125

   • (iv)

     $\frac{1500}{1600} = 0.9375$16001500​=0.9375

   • (vi)

     $\frac{23}{23^2 \cdot 5^2} = 0.04$232⋅5223​=0.04

   • (vii)

     $\frac{129}{2^2 \cdot 5^7} = 0.000625$22⋅57129​=0.000625

   • (viii)

     $\frac{6}{15} = 0.4$156​=0.4

   • (ix)

     $\frac{35}{50} = 0.7$5035​=0.7

   View Solution

3. Question:
   The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form

   $\frac{p}{q}$qp​, what can you say about the prime factors of

   $q$q?
   (i) 43.123456789
   (ii) 0.120120012000120000...
   (iii) 43.123456789... (repeating)

   Answer:

   • (i) 43.123456789 is a non-terminating non-repeating decimal, hence irrational.

   • (ii) 0.120120012000120000... is a repeating decimal, hence rational. The prime factors of

     $q$q must be 2 and 5.

   • (iii) 43.123456789... (repeating) is a repeating decimal, hence rational. The prime factors of

     $q$q must be 2 and 5.

   View Solution

FAQs: Class 10 Maths Chapter 1 – Real Numbers Exercise 1.4

Q1. What is the focus of Exercise 1.4?
Answer:
Exercise 1.4 focuses on finding the greatest common divisor (gcd) of two numbers using Euclid’s Division Algorithm and solving related problems on divisibility and remainders.

Q2. How do I apply Euclid's Division Algorithm in Exercise 1.4?
Answer:
To apply Euclid's Division Algorithm, divide the dividend (a) by the divisor (b) and express it as:
a = b × q + r,
where a is the dividend, b is the divisor, q is the quotient, and r is the remainder. Repeat this process until the remainder is 0, and the divisor at that step is the gcd.

Q3. How do I find the gcd using Euclid's Division Algorithm?
Answer:
Perform the division steps repeatedly until the remainder is 0. The divisor at the last step, where the remainder is 0, is the gcd of the two numbers.

Q4. What are the key concepts needed to solve this exercise?
Answer:
You need to understand divisibility of numbers, quotients, and remainders. The key concept is Euclid’s Division Algorithm and the steps for applying it to find the gcd.

Q5. How do NCERT Solutions help with exam preparation?
Answer:
These solutions provide clear explanations of Euclid’s Division Algorithm, helping students solve problems confidently and understand the concept of gcd. They are an excellent reference for practicing and strengthening the number theory concepts needed for the exam.